Problem: Given that the point $(9,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=\frac{f(2x)}2+2$. What is the sum of coordinates of that point?
Explanation: Since $(9,7)$ is on the graph of $y=f(x)$, we know  \[7=f(9).\]If we substitute $x=\frac92$ into $2y=\frac{f(2x)}2+2$ we get  \[2y=\frac{f(2\cdot9/2)}2+2=\frac72+2=\frac{11}2.\]Therefore $(x,y)=\left(\frac92,\frac{11}4\right)$ is on the graph of  \[2y=\frac{f(2x)}2+2.\]The sum of these coordinates is  \[\frac92+\frac{11}4=\boxed{\frac{29}4}.\]